Here is a proof of a property of a martingale $X$ relative to the filtration $(F_{n})$:
$n\gt m,\\$ $\\ \\ E[X_n|F_m]=E[E[X_n|F_{n-1}]|F_m]=E[X_{n-1}|F_m]=...=E[X_m|F_m]=X_m$
In the definition of a martingale, we are given that $E[X_n|F_{n-1}]=X_{n-1}$.
Can someone explain to me the steps in this proof? In particular how does the first equality hold?
The first equality is using the tower property of conditional expectation ($E[X|G] = E[E[X|F] | G]$ whenever $G \subset F$.
The second equality is using the martingale property of X (i.e. $E[X_n|F_{n-1}] = X_{n-1}$)
The rest of the proof is just by applying induction using the tower property and the martingale property over and over again.